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Overview. Digital Systems, Computers, and Beyond Information Representation Number Systems [binary, octal and hexadecimal] Arithmetic Operations Base Conversion Decimal Codes [BCD (binary coded decimal)] Alphanumeric Codes Parity Bit Gray Codes. Digital Logic.

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Overview

Overview

  • Digital Systems, Computers, and Beyond

  • Information Representation

  • Number Systems [binary, octal and hexadecimal]

  • Arithmetic Operations

  • Base Conversion

  • Decimal Codes [BCD (binary coded decimal)]

  • Alphanumeric Codes

  • Parity Bit

  • Gray Codes


Digital logic

Digital Logic

  • Deals with the design of computers

  • Early computers were used for computations of discrete elements (digital)

  • The term logic is applied to circuits that operate on a set of just two elements with values True and False


D igital c omputer s ystems digital system

DIGITAL & COMPUTER SYSTEMS - Digital System

Discrete

Discrete

Information

Inputs

Discrete

Processing

Outputs

System

System State

  • Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs.


Digital computer example

Digital Computer Example

Memory

Control

Datapath

CPU

unit

Inputs: keyboard, mouse, wireless, microphone

Outputs: LCD screen, wireless, speakers

Input/Output

Synchronous or Asynchronous?


And beyond embedded systems

And Beyond – Embedded Systems

  • Computers as integral parts of other products

  • Examples of embedded computers

    • Microcomputers

    • Microcontrollers

    • Digital signal processors


Embedded systems

Embedded Systems

  • Examples of Embedded Systems Applications

    • Cell phones

    • Automobiles

    • Video games

    • Copiers

    • Dishwashers

    • Flat Panel TVs

    • Global Positioning Systems


I nformation r epresentation signals

INFORMATION REPRESENTATION - Signals

  • Information represents either physical or man-made phenomena

  • Physical quantities such as weight, pressure, and velocity are mostly continuous (may take any value)

  • Man-made variables can be discrete such as business records, alphabet, integers, and unit of currencies.

  • This discrete value can be represented using multiple discrete signals.

  • Two level, or binary signals are the most prevalent values in digital systems. 


Signal examples over time

Signal Examples Over Time

Time

Continuous in value & time

Analog

Digital

Discrete in value & continuous in time

Asynchronous

Discrete in value & time

Synchronous


Signal example physical quantity voltage

Signal Example – Physical Quantity: Voltage

Threshold Region

  • Thetwo discrete values by ranges of voltages called LOW and HIGH

  • Binary values are represented abstractly by:

    • Digits 0 and 1, words (symbols) False (F) and True (T), words (symbols) Low (L) and High (H), and words On and Off.


Binary values other physical quantities

Binary Values: Other Physical Quantities

  • What are other physical quantities represent 0 and 1?

    • CPU Voltage

    • Disk

    • CD

    • Dynamic RAM

Magnetic Field Direction

Surface Pits/Light

Electrical Charge


N umber s ystems representation

NUMBERSYSTEMS – Representation

j = - 1

i = n - 1

å

å

+

i

j

(Number)r

=

A

r

A

r

i

j

i = 0

j = - m

(Integer Portion)

(Fraction Portion)

+

  • Positive radix, positional number systems

  • A number with radixr is represented by a string of digits:An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m +1 A- min which 0 £ Ai < r and . is the radix point.

  • Digits available for any radix r system 0,1,2,…, r-1

  • The string of digits represents the power series:

(

)

(

)


Number systems examples

Number Systems – Examples


Special powers of 2

Special Powers of 2

  • 210 (1024) is Kilo, denoted "K"

  • 220 (1,048,576) is Mega, denoted "M"

  • 230 (1,073, 741,824)is Giga, denoted "G"

  • 240 (1,099,511,627,776) is Tera, denoted “T"


A rithmetic o perations binary arithmetic

ARITHMETIC OPERATIONS - Binary Arithmetic

  • Single Bit Addition with Carry

  • Multiple Bit Addition

  • Single Bit Subtraction with Borrow

  • Multiple Bit Subtraction

  • Multiplication

  • BCD Addition


Single bit binary addition with carry

Single Bit Binary Addition with Carry


Multiple bit binary addition

Multiple Bit Binary Addition

  • Extending this to two multiple bit examples:

    Carries 00

    Augend 01100 10110

    Addend +10001+10111

    Sum

  • Note: The 0 is the default Carry-In to the least significant bit.


Single bit binary subtraction with borrow

Z

Z

1

0

1

0

1

0

0

1

Single Bit Binary Subtraction with Borrow

X

X

0

0

0

0

1

1

1

1

BS

BS

11

0 0

1 0

1 1

0 1

0 0

1 1

0 0

- Y

- Y

-0

-0

-1

-1

-0

-0

-1

-1

  • Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B):

  • Borrow in (Z) of 0:

  • Borrow in (Z) of 1:


Multiple bit binary subtraction

Multiple Bit Binary Subtraction

  • Extending this to two multiple bit examples:

    Borrows00

    Minuend 10110 10110

    Subtrahend- 10010- 10011

    Difference

  • Notes: The 0 is a Borrow-In to the least significant bit. If the Subtrahend > the Minuend, interchange and append a – to the result.


Binary multiplication

Binary Multiplication


B ase c onversion positive powers of 2

BASE CONVERSION - Positive Powers of 2

  • Useful for Base Conversion

Exponent

Value

Exponent

Value

0

1

11

2,048

1

2

12

4,096

2

4

13

8,192

3

8

14

16,384

4

16

15

32,768

5

32

16

65,536

6

64

17

131,072

7

128

18

262,144

8

256

19

524,288

9

512

20

1,048,576

10

1024

21

2,097,152


Converting binary to decimal

Converting Binary to Decimal

  • To convert to decimal, use decimal arithmetic to form S (digit × respective power of 2).

  • Example:Convert 110102to N10:


Commonly occurring bases

Commonly Occurring Bases

Name

Radix

Digits

Binary

2

0,1

Octal

8

0,1,2,3,4,5,6,7

Decimal

10

0,1,2,3,4,5,6,7,8,9

Hexadecimal

16

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

  • The six letters (in addition to the 10

    integers) in hexadecimal represent:


Numbers in different bases

Numbers in Different Bases

  • Good idea to memorize!

Decimal

Binary

Octal

Hexa

decimal

(Base 10)

(Base 2)

(Base 8)

(Base 16)

00

00000

00

00

01

00001

01

01

02

00010

02

02

03

00011

03

03

04

00100

04

04

05

00101

05

05

06

00110

06

06

07

00111

07

07

08

01000

10

08

09

01001

11

09

10

01010

12

0A

11

0101

1

13

0B

12

01100

14

0C

13

01101

15

0D

14

01110

16

0E

15

01111

17

0F

16

10000

20

10


Conversion between bases

Conversion Between Bases

  • To convert from one base to another:

1) Convert the Integer Part

2) Convert the Fraction Part

3) Join the two results with a radix point


Conversion details

Conversion Details

  • To Convert the Integral Part:

    Repeatedly divide the number by the new radix and save the remainders. The digits for the new radix are the remainders in reverse order of their computation. If the new radix is > 10, then convert all remainders > 10 to digits A, B, …

  • To Convert the Fractional Part:

    Repeatedly multiply the fraction by the new radix and save the integer digits that result. The digits for the new radix are the integer digits in order of their computation.If the new radix is > 10, then convert all integers > 10 to digits A, B, …


Example convert 46 6875 10 to base 2

Example: Convert 46.687510 To Base 2

  • Convert 46 to Base 2

  • Convert 0.6875 to Base 2:

  • Join the results together with the radix point:


Additional issue fractional part

Additional Issue - Fractional Part

  • Note that in this conversion, the fractional part can become 0 as a result of the repeated multiplications.

  • In general, it may take many bits to get this to happen or it may never happen.

  • Example Problem: Convert 0.6510 to N2

    • 0.65 = 0.1010011001001 …

    • The fractional part begins repeating every 4 steps yielding repeating 1001 forever!

  • Solution: Specify number of bits to right of radix point and round or truncate to this number.


Checking the conversion

Checking the Conversion

  • To convert back, sum the digits times their respective powers of r. 

  • From the prior conversion of  46.687510

    1011102 = 1·32 + 0·16 +1·8 +1·4 + 1·2 +0·1

    = 32 + 8 + 4 + 2

    = 46

    0.10112 = 1/2 + 1/8 + 1/16

    = 0.5000 + 0.1250 + 0.0625

    = 0.6875


Octal hexadecimal to binary and back

Octal (Hexadecimal) to Binary and Back

  • Octal (Hexadecimal) to Binary:

    • Restate the octal (hexadecimal) as three (four) binary digits starting at the radix point and going both ways.

  • Binary to Octal (Hexadecimal):

    • Group the binary digits into three (four) bit groups starting at the radix point and going both ways, padding with zeros as needed in the fractional part.

    • Convert each group of three bits to an octal (hexadecimal) digit.


Octal to hexadecimal via binary

Octal to Hexadecimal via Binary

  • Convert octal to binary.

  • Use groups of four bits and convert as above to hexadecimal digits.

  • Example: Octal to Binary to Hexadecimal

    6 3 5 . 1 7 7 8

  • Why do these conversions work?


Examples

Examples

  • Convert the following numbers to the specified bases

  • (F3D2)16 (?)10

  • (341)8  (?)16

  • (153)10  (?)2

  • (1001101)2  (?)10

  • (245)10  (?)16


Binary numbers and binary coding

Binary Numbers and Binary Coding

  • Flexibility of representation

    • Within constraints below, can assign any binary combination (called a code word) to any data as long as data is uniquely encoded.

  • Information Types

    • Numeric

      • Must represent range of data needed

      • Very desirable to represent data such that simple, straightforward computation for common arithmetic operations permitted

      • Tight relation to binary numbers

    • Non-numeric

      • Greater flexibility since arithmetic operations not applied.

      • Not tied to binary numbers


Non numeric binary codes

Non-numeric Binary Codes

Binary Number

000

001

010

011

101

110

111

  • Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers.

  • Example: Abinary codefor the sevencolors of therainbow

  • Code 100 is not used

Color

Red

Orange

Yellow

Green

Blue

Indigo

Violet


Number of bits required for coding

Number of Bits Required For coding

  • Given M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships:

    2n³M> 2(n – 1)

    n = log2M where x , called the ceilingfunction, is the integer greater than or equal to x.

  • Example: How many bits are required to represent decimal digits with a binary code?


Number of elements represented

Number of Elements Represented

  • Given n digits in radix r, there are rn distinct elements that can be represented.

  • But, you can represent m elements, m < rn

  • Examples:

    • You can represent 4 elements in radix r = 2 with n = 2 digits: (00, 01, 10, 11).

    • You can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000).

    • This second code is called a "one hot" code.


Bcd code

BCD Code

  • Binary Coded Decimal

  • Decimal digits stored in binary

    • Four bits/digit

    • Like hex, except stops at 9

    • Example

      931 is coded as 1001 0011 0001

    • Easier to interpret, but harder to manipulate.

  • Remember: these are just encodings. Meanings are assigned by us.


A lphanumeric c odes ascii character codes

ALPHANUMERIC CODES - ASCII Character Codes

  • American Standard Code for Information Interchange

  • This code is a popular code used to represent information sent as character-based data. It uses 7-bits to represent:

    • 94 Graphic printing characters.

    • 34 Non-printing characters

  • Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return)

  • Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).

(Refer to Table 1

-4 in the text)


Ascii properties

ASCII Properties

ASCII has some interesting properties:

  • Digits 0 to 9 span Hexadecimal values 3016

to 3916

.

  • Upper case A

-

Z span 4116

to 5A16

.

  • Lower case a

-

z span 6116

to 7A16

.

  • Lower to upper case translation (and vice versa)

occurs by

flipping bit 6.

  • Delete (DEL) is all bits set,

a carryover from when

punched paper tape was used to store messages.

  • Punching all holes in a row erased a mistake!


P arity b it error detection codes

PARITY BIT Error-Detection Codes

  • Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.

  • A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.

  • A code word has even parity if the number of 1’s in the code word is even.

  • A code word has odd parity if the number of 1’s in the code word is odd.


4 bit parity code example

4-Bit Parity Code Example

  • Fill in the even and odd parity bits:

  • The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data.

Even Parity

Odd Parity

Message

Parity

Parity

Message

-

-

000

000

-

-

001

001

-

-

010

010

-

-

011

011

-

-

100

100

-

-

101

101

-

-

110

110

-

-

111

111

-

-


G ray c ode decimal

GRAYCODE – Decimal

  • What special property does the Gray code have in relation to adjacent decimal digits?

Decimal

BCD

Gray

0

0000

0000

1

0001

0100

2

0010

0101

3

0011

0111

4

0100

0110

5

0101

0010

6

0110

0011

7

0111

0001

8

1000

1001

9

1001

1000


Optical shaft encoder

Optical Shaft Encoder

  • Does this special Gray code property have any value?

  • An Example: Optical Shaft Encoder

111

000

100

000

B

0

B

1

101

001

001

110

B

2

G

2

G

1

G

111

0

010

011

101

100

011

110

010

(b) Gray Code for Positions 0 through 7

(a) Binary Code for Positions 0 through 7


Shaft encoder continued

Shaft Encoder (Continued)

  • How does the shaft encoder work?

  • For the binary code, what codes may be produced if the shaft position lies between codes for 3 and 4 (011 and 100)?

  • Is this a problem?


Shaft encoder continued1

Shaft Encoder (Continued)

  • For the Gray code, what codes may be produced if the shaft position lies between codes for 3 and 4 (010 and 110)?

  • Is this a problem?

  • Does the Gray code function correctly for these borderline shaft positions for all cases encountered in octal counting?


Unicode

UNICODE

  • UNICODE extends ASCII to 65,536 universal characters codes

    • For encoding characters in world languages

    • Available in many modern applications

    • 2 byte (16-bit) code words

    • See Reading Supplement – Unicode on the Companion Website http://www.prenhall.com/mano


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